Lemma 5.5 ([1]} 2.9) 
Let \(f: X \rightarrow Y\) , \(g: W \rightarrow Y\)  be log morphisms between fs log stacks. Suppose further that \(W\)  is free, that is, the stalks \(\overline{\mathcal {M}}_{W,w}\)  are free monoids for all \(w \in W\) . Then \(g\)  is transverse to \(f\)  if for any geometric point \(x \in X\) , \(y \in Y\)  and \(w \in W\)  with \(f(x) = y = g(w)\) , and for any face \(F\)  of \(\sigma _x\)  we have that \(\Sigma (g)^{-1}(\Sigma (f)(F)) \subset \sigma _w\)  is a face of \(\sigma _w\) .

